Periodic Sequences a periodic sequence with integer period n xn=xn+n for all n Acos(won+o)=Acoslwon+woN+) WoN=2T k, where k is integer N=2nk/w where k is integer 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ
17 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Periodic Sequences uA periodic sequence with integer period N x[n] x[n N] for all n A w n A w n w N 0 0 0 cos cos 2 , integer 0 w N k where k is 0 N 2 k / w , where k is integer
X 2.1 Examples of Periodic Sequences x [n]=cos(n/4) Suppose it is periodic sequence with period N x[n]=x[n+ni cos(n/4)=cos((n+N)/4 Tn/4+2T k=t n/4+NI/4, k: integer N=2丌k/(丌/4)=8k k=1,→>N=8=2x/W 18 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ
18 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.1 Examples of Periodic Sequences uSuppose it is periodic sequence with period N [ ] [ ] x1 n x1 n N [ ] cos / 4 x1 n n cos n / 4 cos n N/ 4 n / 4 2 k n / 4 N / 4, k :integer 0 k 1, N 8 2 / w N 2 k / ( / 4) 8k
EX. 2.1 Examples of periodic sequences 2丌3兀 x[n]=cos(3zn/8) A Suppose it is periodic sequence with period n rn]=xn+n cos(3n/8=cos 3(n+N/ 3兀n/8+2丌k=3丌n/8+3N丌/8,k: Integer N=2k/w=2/(3z/8)k=3,→>N=16 N=2x3/W≠2丌/W( for continuous signal) 19 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ
19 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. uSuppose it is periodic sequence with period N [ ] [ ] x1 n x1 n N cos3 n / 8 cos3 n N/ 8 3 n / 8 2 k 3 n / 8 3N / 8, k :integer k 3, N 16 EX. 2.1 Examples of Periodic Sequences [ ] cos3 /8 x1 n n 8 3 8 2 0 N 2 k / w 2 k / (3 / 8) 0 0 N 2 3/ w 2 / w ( for continuous signal)
EX 2.1 Non-Periodic Sequences xIn=cost n Suppose it is periodic sequence with period N x2n=x,ln+n cos(n =cos(n+ N for n+2T k=n+N, k: integer, there is no integer N 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ
20 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.1 Non-Periodic Sequences uSuppose it is periodic sequence with period N [ ] [ ] x2 n x2 n N x [n] cos n 2 cos n cos(n N) 2 , :integer, integer for n k n N k there is no N
High and low Frequencies in Discrete-time signal xn= a cos(won) (a)W=0or2π Frequency: The rate at which a repeating event occurs. (b)W=π8or15/8 (c)=πor7/4 Zhongguo Liu_Biomedical Engineering_ Shandong Univ
21 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. High and Low Frequencies in Discrete-time signal 0 x[n] Acos(w n) (b) w0 = /8 or 15/8 (c) w0 = /4 or 7/4 (d) w0 = Frequency: The rate at which a repeating event occurs. (a) w0 = 0 or 2